Abstract

Abstract
Recently, there has been increasing interest in investigating the interrelationships among
the component stochastic processes of a dynamical system. The applications of these studies are to be found in various fields such as Economics, Neuroscience, Engineering and Neurobiology. Also the determination of the direction of the information flow is one of the important subjects studied widely. These investigations have usually been implemented in the time and frequency domains. Consequently, several mathematical and statistical
procedures have been developed to facilitate these analyses.
The aim of this thesis is to discuss the relationships between stochastic processes of a relatively short time duration. Specifically, the research concerns the analysis of the electrical activity of the dysfunctional brain, where the available data belong to a right-handed focal epileptic patient. EEG signals are recorded directly from the scalp using numbered electrodes according to the International 10/20 system introduced by Jasper [1958]. The analysis is only performed for processes of the left hemisphere as they represent the dominant hemisphere. Moreover, since each region of the brain is responsible
for a special function, we have chosen five processes to represent the five main lobes of the brain; the frontal lobe, the central region, the parietal lobe, the occipital lobe and the temporal lobe.
The analyses of these signals are carried out using four spectral density estimation procedures, namely the multivariate autoregressive model of order 2; the average of periodograms of adjacent segments of the single record; the smoothed periodogram approach for the entire record; and the multi-taper method. Thereafter comparisons among the
results of these methods are made. The strength of the correlation between signals is measured by coherence and partial coherence functions. Also, the Granger causality concept is implemented for these data in the form of determining the direction of the information flow between these signals using the partial directed coherence (PDC) proposed by Baccala and Sameshima [2001] using the statistical level of significance suggested by Schelter et
al.[2005]. The structure of the causal influences produced by the PDC shows that there are statistically significant reciprocal causal effects between processes representing the
brain's region, the frontal lobe, the central area, the parietal lobe and the temporal lobe. However, there are two uni-directed causal influence relations, one is between the central area and the occipital lobe and the second one is between the occipital and temporal lobes. The indirect causal influences are detected between these processes throughout the process representing the temporal lobe. Generally, the values of the PDC in the anterior-posterior
direction are larger than the values of the PDC in the opposite direction. Also, the causal influences of each process on the temporal lobe process is larger than the causal influences in the opposite direction.
The spectral analyses show that the estimated power spectra and coherences of these signals are approximately peak in the delta wave band of frequency [1, 4) Hz. The significant non-zero estimated coherences are captured between the brain's lobes except for the occipital lobe which is uncorrelated with any of the other lobes. The depth of non-zero significant estimated coherences is given by partial coherence, which measures the strength of
the estimated coherence between any two processes after removing the linear influence of one or more other processes. For the current data, we found that the depth of correlations depends on the spectral estimation method adopted. For example, the depth of correlation
is of order 2 for the method of averaging across periodograms of adjacent segments of the
single record and the method of smoothed periodogram of the entire single record and is of order one for the multi-taper method. However, the depth of correlations is unknown for the multivariate autoregressive model of order 2. The comparisons made between the results
of the four spectral estimation methods mentioned previously, indicated that MVAR is not sensitive to rapid changes occurring in the signal such as the effect of the notch filter at 60Hz and a calibration signal at 47Hz, while the other three methods exhibited
good sensitivity to these changes with different strengths of responses. Furthermore, the smoothed periodogram and the multi-taper methods persistently detect the notch filter
effect at 60Hz in the ordinary estimated coherence curves, while the method of averaging across periodograms of adjacent segments of the single record does not.